GEOGEBRA AND NUMERICAL REPRESENTATIONS: A PROPOSAL
INVOLVING FUNDAMENTAL THEOREM OF ARITHMETIC
Gerson Pastre de Oliveira
Pontifícia Universidade Católica de São Paulo and Universidade Paulista (Brasil);
[email protected]
This paper reports a qualitative research whose subjects were Elementary School Teachers who
took part in a workshop about primality of positive integers and the Fundamental Theorem of
Arithmetic (FTA). These topics was dealt with from different technological perspectives and
analysed under a theoretical proposal connected to the concepts of transparency and opacity of
numerical representations and to the "humans-with-media" approach. The interactions occurred in
a Post Graduate Program in Mathematics Education and they consisted of two activities created to
ask subjects which numbers in a random list would be prime. The analysis showed that participants
had difficulties with FTA, which led them to adopt strategies with a high cognitive cost and make
mistakes. Likewise, data showed that hindrances were overcome based on the educational proposal
planned from a configuration of humans-with-GeoGebra.
Keywords: fundamental theorem of arithmetic; numerical representations; digital technologies in
education; human-with-media; GeoGebra.
INTRODUCTION
To recognize a positive integer as a prime or a composite number may seem a simple task, not very
interesting in terms of teaching or researching in the field of Mathematics Education. Once stated
that a natural number is prime if, and only if, it can be divided uniquely by itself and by one, we
might think that there are no associated complexities. Ultimately, it just seems to be a question of
finding a divisor between 2 and the square root of the number tested for primality; or of applying the
so-called „divisibility rules‟. Such a search, however, can have quite a high operational cost. For
example, if the candidate number is 30847, we must consider that its first divisor is 109 and that
there is only another one apart from the number tested for primality itself and number 1, which is
283.
In order to deal with such problem, this work describes a research whose participants were a group
of basic education teachers from public schools, involved in the projects “Mathematics Teaching
and Learning Processes in Technological Environments”, constituted through a partnership between
the Pontifical Catholic Universities (PUC) of Sao Paulo (Brazil) and of Lima (Peru), and
“Technologies and Mathematical Education: researches on fluency in devices, tools, artefacts and
interfaces”, also linked to the PUC Sao Paulo [1]. In the proposed activities, teachers had to identify
whether several numbers were prime, in situations where divisibility rules, for example, were an
inefficient strategy and where FTA knowledge would be relevant. Through the chosen research
instruments – a question about the subject „primality of natural numbers‟ and an application built by
using the software GeoGebra – we sought to highlight the strategies used by the subjects to solve
the problems, their ideas about the representation of natural numbers and the influence of the type of
technology on the mobilization of the mathematical knowledge at issue. In this respect, arguments
related to the number representation arise, as well as arguments related to the use of technology in
Mathematics Education, which leads to the following theoretical treatment.
ICTMT 13
Lyon
1
NUMERICAL REPRESENTATIONS
One of the core concepts addressed in this research refers to transparency and opacity of numerical
representations. In this respect, the study of Zazkis and Liljedahl (2004) mentions the role of
representations in the field of natural numbers. In their paper, the authors discussed data obtained
from a research also conducted with basic education teachers in training, which focused on their
understanding of prime numbers, so as to detect the factors that interfere with such understanding.
The argument used in the analysis of collected data is that the lack of transparency of prime
numbers representation is a hindrance to understand them.
This idea was taken from the paper of Lesh, Behr and Post (1987). When referring to different
representations of rational numbers, the authors show that they “incorporate” mathematical
structures, meaning that they represent them materially. Thus, representational systems can be
regarded as opaque or transparent: a transparent representation has no more nor less meaning than
the ideas or structures it represents, while an opaque representation emphasizes some aspects of the
ideas or structures it represents while hiding some other ones. When having different
representational possibilities, a didactic strategy should, for example, capitalise on the strengths of a
specific representational system and minimize its weaknesses.
From Lesh, Behr and Post‟s proposal (1987), Zazkis and Gadowsky (2001) introduce the notion of
relative transparency and opaqueness, focusing on numerical representations. The authors suggest,
on their paper, “that all representations of numbers are opaque in the sense that they always hide
some of the features of a number, although they might reveal other, with respect to which they
would be „transparent‟”. As an example, the authors provide a list with the following items: (a)
2162, (b) 363, (c) 3 x 15552, (d) 5 x 7 x 31 x 43 + 1, (e) 12 x 3000 + 12 x 888. The authors mention
that such expressions do not seem to represent the same number, 46656, pointing that each
representation shifts our attention to different properties of it.
Based on the above-mentioned concepts, the authors claim that all numeric representations are
opaque, but they have transparent features. In relation to the work we describe here, activities
involving the features of numerical representations were presented to the subjects by using different
means and technological interfaces.
ABOUT THE USE OF TECHNOLOGIES IN MATHEMATICS EDUCATION
Technologies have generally been present in education processes from time immemorial, if we
consider, like Lévy (1993) that transmutation of temporalities in Human History brought about
different instruments; some of them prevailed over the others depending on the evolutionary
character of the society in a given time. Thus, orality, writing and information technology are
enrolled as technologies of intelligence. The ascendancy of information technology did not suppress
previous technological proposals, but constituted, in relation to them, a feature of redefining
functions and, in the last resort, of convergence. Thus, it is unavoidable to associate the use of some
technology in processes of teaching or learning: a construction of knowledge and its forms of
access, therefore, were always linked to more or less material tools of technological nature.
Therefore, the perspective shown so far gets people and technologies involved in the construction of
mathematical knowledge. However, technologies do not play a part in such association as humancapacity substitutes, not even as a supplement to them, but as reorganizers of human thinking
(Tikhomirov, 1981). In this sense, computing applications for instance allows unusual mediation
forms, by delegating to the computer the role of tool for the human mental activity, detainer of
ICTMT 13
Lyon
2
similar functions to those carried out by language in the vygotskinian logics. Similar reasoning can
be applied to contemporary devices, like tablets or cell phones.
Such considerations support the statement that, in Mathematics, learning is a process involving
technologies which are somehow integrated in people; this allows intentionalities, strategies,
plannings and wills to come into play. According to Borba and Villarreal (2005), such integration
must be of such an order that it excludes any attempt to see these items, people and technologies, as
separate groups. Therefore, for these authors, mathematical knowledge is formed from a collective
of humans-with-media, considering that media reorganize people‟s thinking and that the presence of
different technologies conditions the production of different forms of knowledge. Therefore, in the
research here explained, the two activities described below tried to investigate the understanding of
numerical representations related to prime numbers and the FTA presented by a group of teachers in
continuous training, based on the mobilization of these collective of people-with-technologies at
different moments, with different media: during the first activity, teachers-with-pencil-and-paper; in
the second, teachers-with-computers-and-GeoGebra.
METHODOLOGICAL APPROACH
The participants of this qualitative research are eight teachers from basic education public schools in
São Paulo and Pará, all of them voluntary in workshops carried out in the framework of the projects
mentioned at the start of this text. The research we describe here was conducted in one of the
computing labs of the abovementioned institution, in a single session which lasted around four
hours. Among the subjects so described, five are in basic and secondary education while three are
only in basic education. Moreover, all of them have completed their bachelor‟s degree in
Mathematics, but three of them are attending an Academic Master in Mathematics Education while
two have completed their Specialization in Mathematics Education.
Participants in this study were invited to working with two kinds of activities involving knowledge
on primality in the framework of the Theory of Numbers. The first activity had a question, to be
solved individually, by presenting the answer in writing. The formulation is as follows: “Consider F
= 151 x 157. Is F a prime number? Circle YES/NO, and explain your decision” (Zazkis & Liljedahl,
2004, p.169). To answer to the abovementioned question, students would have to write down the
option „No‟, since the representation in question, with transparent features, shows that F is
composite, and it even relates the component prime factors. Students should resort to the
Fundamental Theorem of Arithmetic (FTA) to conclude that the said decomposition is unique, apart
from the order.
Otherwise, subjects could perform the multiplication contained in the question, obtaining 23707, an
opaque representation regarding detection of the composite character of the number. From this other
representation, although it is not necessary in order to resolve the problem, they could make tests
with several possible divisors. It can even happen that the subject tries to divide 23707 by the prime
numbers from 3 onwards, giving up after some attempts, since natural numbers below 151 are not
divisors of 23707; in this case, participant can even get to wrongly state that the number is prime.
The next activity was carried out immediately after the first one. Individually, the eight teachers had
in front of them a GeoGebra screen containing only one button with the word “Números”
(“Numbers”) on it. They were all informed that the application would raffle nine numbers when
they pressed the said button. Then, researcher instructed subjects with regard to the result they
would see at clicking the button: nine numbers would be shown on the Algebra View of the
software. They had to state, by writing on a blank paper that had been handed out and in a lapse of
ICTMT 13
Lyon
3
20 minutes, whether each number was prime or not. In the meantime, they should not click on the
button with the label “Coisa” (“Do something”) [2] on it (Figure 1) shown after the raffle.
Figure 1. Screen with raffled numbers (developed by the author)
Regarding numbers, it is worth noticing that the construction of the code in javascript, which
carried out the raffling, took into account the random selection among numbers that, in such
context, could be considered „big‟, given that each one of them would be within the interval 1001 to
99999 (Figure 1). The goal was to restrict direct application of the divisibility rules and of division
algorithms. Regarding subjects‟ responses, although numbers could be considered “big”, some of them
were expected to be searched through the application of divisibility rules and/or algorithms of division
by prime numbers, and that such procedures might bring success in some cases (as 47301, which is
divisible by 3) and failure in some other (as 39203, which is not prime, but whose prime factors are 197
and 199). In theory, representations provided by the software were opaque regarding numbers primality,
at least, up to this point of the experiment.
Once the time allowed was over, regardless of the amount of resolutions among the nine proposals,
subjects were invited to click on button “Coisa”. After this action, the Algebra View of GeoGebra
showed the decomposition of each of the nine numbers in prime factors, in the form of lists (Figure
2) – in the case of primes just the number itself was presented.
Figure 2. Numbers and their compositions in prime factors (developed by the author)
Immediately after and still individually, participants were invited to revisit their previous answers in
the light of the new data obtained with the factorization carried out in GeoGebra. At that occasion,
subjects were expected to relate the lists obtained with the numbers raffled at the beginning and,
when they observed their decomposition in prime factors, they should use the FTA so as to decide
on the primality or not of each of the numbers. Ten minutes were allowed for this part of the
activity, after which there would be a debate involving the subjects and the researcher. The button
ICTMT 13
Lyon
4
“Apagar Listas” (“Erase all”) (Figure 2) could be used to go back to the initial screen, which would
allow repeating the experience as many times as considered necessary by participants.
ANALYSIS
At the very beginning of first activity, Teach2 after trying some operations with pencil and paper
stated that 23707 would „probably‟ be prime. Other four participants also stated, wrongly, that the
number in question would be prime, using similar strategies:

Teach3 does countless division operations and ends up stating that “23707 is prime,
because it can be divided by itself and by one”. In this way, he shows he is not aware that
this criterion does not distinguish between prime and composite numbers;

Teach4, after several attempts using division operations, concluded that 23707 would be
a prime number because it “ends in 7 and 7 is prime”;

For Teach5, as the divisibility tests by 2, 3, 5, 7, 11 and 13 „failed‟ (divisions had a
remainder other than zero), the number in question would be prime – in this case the
teacher indicates he believes that “decomposition into prime factors means
decomposition into small prime factors” (Zazkis & Campbell, 1996, p. 215);

In the case of Teach8, number 23707 “is prime, because it is an odd number and it cannot
be divided by its square root or any other prime number”. Like Teach5, Teach8 limited
the universe of prime numbers to the interval within 2 and 13 and it evidenced several
confusions involving the concepts of perfect square numbers and odd numbers.
The representation of F provided in the question formulation has transparent features in relation to
primality, because it presents the number by means of its unique decomposition in prime factors.
However, the abovementioned teachers did not use this idea as expressed in the FTA, which points
out the fact that a numeric representation with mathematical features that make it transparent can be
kept opaque when knowledge about it is not mobilized by the individual. The same authors, apart
from Lesh, Behr and Post (1987) and Zazkis and Gadowsky (2001), state that didactic strategies can
be used in order to reinforces transparent features of a given representational system.
Furthermore, some participants correctly stated that F would not be prime. According to Teach1, “F
is divisible by 151 and 157, which makes it non-prime”. Participants Teach6 and Teach7 said, in a
similar way, that F had other divisors besides itself and 1, which would make it non-prime.
Nevertheless, none of the three participants that answered correctly did show any sign of using the
FTA in their conjectures: when questioned about the possibility of F having other divisors apart
from the ones mentioned, the three of them said it was possible but that they would have to test
numbers up to a certain limit (according to Teach1 up to the number square root; according to
Teach6 and Teach7, up to half the number). Another aspect worth highlighting is that none of the
teachers showed awareness of the fact that factors 151 and 157 represented prime numbers.
From a different perspective, the technological component of the collective humans-with-pen-andpaper, even if intensively used, does not seem to support cognitive movements related to a change
of strategies – in the case of participants who dried up attempts with divisibility algorithms – or to
the use of formal notions in Mathematics, such as the FTA – in the case of participants who stated
that 23707 might have other divisors.
With regard to the second activity, carried out in GeoGebra, participants accessed the application
which was available on the computers in the institution‟s lab and, without further questions, they
ICTMT 13
Lyon
5
chose the raffling of nine numbers, as shown on figure 1. From this moment, teachers had 20
minutes to decide which numbers would be primes. All participants alleged, initially, that time was
too tight and that numbers would be too big (odd numbers between 1001 and 99999) to be able to
provide an answer. The researcher said that they should also state as many results as possible in this
lapse of time, which could not be extended.
Until that moment, the technological aspect in the collective teachers-with-GeoGebra did not have
great influence on the issue of transparency of the numeric representation regarding primality,
because the program interface in question only provided odd random numbers within the said limits.
Thus, as raffled numbers were potentially different, the amount of correct or wrong results showed
significant differences among the subjects. Those whose raffled numbers allowed the application of
divisibility rules or tests with „small‟ prime factors (3 to 13) could be more directly applied obtained
more hits than those whose raffled numbers were 11741 (59 x 199) or 31753 (113 x 281), for
example. Generally, such numbers were wrongly stated to be prime. Even when prime numbers
were identified, as 17231, a doubt used to remain, as in the case of Teach5, who wrote, next to the
said number, “I think it is prime. I tested up to 13”. As we have already seen, this strategy, in the
case of numbers whose prime factors are all higher than 13, is not efficient.
Among the teachers‟ talks during the 20 minutes allowed for the exercise, several references were
made to the numbers „difficult form‟, a clear attempt to refer to their representation, which is clearly
opaque in terms of the feature „primality‟. Teach6 even got to question the goal of GeoGebra in that
context, as the application did not seem, according to him, “to make things easier for those who tried to
solve the problem”.
Before the perplexity caused by the proposal, the researcher, after the time allowed, proceeded to
coordinating a debate with the participants, whose main motivation was raising the conjectures and
strategies that subjects had proposed in order to verify the primality of their nine numbers. None of
the participants said to have tried nor even thought about obtaining the factorization of the numbers
to be tested for primality in order to using knowledge on FTA. Once the debate was ended, the
researcher said that subjects could click on button „Coisa‟, which would show, for each raffled
number, the respective list of component prime factors (figure 2). In 10 minutes, then, teachers had
to review their answers. Immediately after clicking on the button, teachers had to construe the data
showed on the screen.
When they realized the numeric equality and the correspondence between the lists provided and the
raffled numbers (list1 referred to number a1, list2 referred to number a2, and so on), most teachers
started searching for relations among the said components:
Teach6:
Professor, I would like to review my answers.
Researcher:
Yes, why so?
Teach6:
Because I realized something that I had not seen before… the lists… they are the
factors of each number...
Teach1:
By multiplying the numbers in the lists we obtain the raffled numbers…
Teach4:
True, but there are cases where only one number appears… these numbers are
prime, as we can only multiply by one!
Teach3:
[does not seem convinced] Professor, I´ll raffle the numbers again… [after
repeating the raffling and factorization] It‟s true! Prime numbers do not have
factors, only composite numbers do.
ICTMT 13
Lyon
6
Teach4:
A prime number‟s factors are itself and one...
Teach7:
I was thinking... In my case, one of the numbers is 88739... Factorization is shown
as 7, 7 and 1811. I could as well write 49 and 1811, right?
Teach8:
[after some discussion with pairs] I think that 49 can appear, but 49 is not prime,
and lists show prime factors of numbers. The idea is that only prime numbers
appear. We can see that all the factors, in all the numbers, are prime.
Teach4:
You are right. Any number can be written as a product of prime factors! This is it!
Wow! First question was obvious! There is only one decomposition into prime
numbers for each number.
Teach8:
It is the Fundamental Theorem of Arithmetic!
After these observations, subjects could tell which numbers were prime and which were not, by
repeating raffling and the whole procedure several times. As noted by Borba and Villarreal (2005),
visualization and experimentation were important factors in the new strategy adopted by subjects
from configuration humans-with-GeoGebra. According to these authors, such items allow, among
other actions, for example, to invest in generating conjectures about the problems at issue (and
testing them through countless examples), bring to light some results which were not known before
the experiments and testing different ways of collecting results. The access to visual components, in
the consolidation of the results of actions carried out by people-with-GeoGebra, became a way to
transform the understanding they had about the problems at issue.
Another aspect that must be taken into account in the configuration teachers-with-GeoGebra, is the
dynamism of digital technologies, that has been seen here as a possibility to manipulate the
parameters, attributes or values which served the constitution and/or definition of a mathematical
construct in a computerized context. Before the possibilities open by this resource, a fundamental
investigative movement to mathematics finds consistent subsidies from the development, testing
and validation (or refutation) of conjectures. This could be largely seen in the experiment we
describe here, when teachers invested, through experimenting and visualizing, in the procedure
repetition, using the regularities observed in factorizations – and dynamically obtained, as a means
to support the reorganization of ideas on the primality of the presented numbers. All these factors
collaborated to mobilise knowledge on FTA for the solving of the problem.
FINAL CONSIDERATIONS
The discussion following the last activity was quite fruitful: participants stated that, in activity 1
they had not realized that the „way‟ in which the number was written (its representation) allowed the
question to be answered directly by mobilizing knowledge related to the FTA.
Regarding activity 2, participants said that numbers were not expressed in a „convenient way‟
(transparent representation), i.e., according to them, they were „big numbers‟ that were not
decomposed in prime factors. Teachers mentioned the fact that they had spent the 20 minutes to try
and say which of the numbers were prime, but that, if they had the appropriate representation in
factors and had remembered the FTA, they would have done this much faster. This last feature was
realized by them when they clicked on the second button, causing the decomposition of the numbers
in prime factors to appear. The participants concluded that, when there were other factors besides 1
and the number itself (trivial factorization), the number in question would not be prime. Moreover,
teachers highlighted the importance of FTA knowledge and of the use of GeoGebra in the
procedures, saying that this would be a good way to approach the subject in the classroom.
ICTMT 13
Lyon
7
Moreover, in this context, the Geogebra´s application would work as a „calculator to decompose in
prime factors‟, turning a numeric representation, opaque in relation to primality, into a transparent
one, provided that the knowledge about the FTA has been mobilized. In this case, the configuration
people-with-GeoGebra contributed in a more efficient way to direct the problem-solving effort
towards a strategy that brings more chances of success. Dialogues show, although only to some
extent, the renegotiation of meanings, the conjectural reformulations and the reorganization of
thinking which allowed the right answer to come out. Another feature that must be highlighted is
that representations of prime numbers can give opportunity to transparent features, as soon as the
appreciation of underlying meanings and concepts is taken into account. When this does not occur,
the tendency is to call for large, expensive solutions in cognitive terms. To investigate the nature of
these processes and develop proposals in order to avoid such difficulties to remain among basic
school teachers seems to be an important challenge, open to new researches.
NOTES
1. Research projects supported by FAPESP and CNPq, respectively (Brazilian scientific development agencies).
2. The button title could then be „Factorize‟, but the idea was to promote a research where subjects were the authors of
the hypothesis formulated to solve the problem: the use of this title might imply that teachers needed, compulsorily, to
do the factorization of the numbers, which would compromise their autonomy.
REFERENCES
Borba, M. C, & Villarreal, M. (2005). Humans-with-media and the reorganization of mathematical
thinking: Information and communication technologies, modelling, visualization and
experimentation. New York: Springer.
Lesh, R., Behr, M., & Post, T. (1987). Rational Number Relations and Proportions. In C. Janiver
(Ed.). Problems of Representations in the Teaching and Learning of Mathematics (pp. 41-58).
Hillsdale: Lawrence Erlbaum.
Lévy, P. (1993). As tecnologias da inteligência: O futuro do pensamento na era da informática. Rio
de Janeiro: Editora 34.
Tikhomirov, O. K. (1981). The psychological consequences of computerization. In Wertsch, J. V.
(Ed.). The concept of activity in soviet psychology (pp. 256–278). New York: M. E. Sharpe.
Zazkis, R., & Campbell, S. R. (2006). Number Theory in mathematics education research. In
Zazkis, R.; Campbell, S. (Eds.). Number Theory in mathematics education: perspectives and
prospects (pp. 1-18). New Jersey: Lawrence Erlbaum Press.
Zazkis, R., & Campbell. S. R. (1996). Prime decomposition: Understanding uniqueness. Journal of
Mathematical Behavior, 15, 207-218.
Zazkis, R., & Gadowsky, K. (2001). Attending to transparent features of opaque representations of
natural numbers. In Cuoco, A. (Ed.). NCTM 2001 Yearbook: The roles of representation in school
mathematics (pp. 41-52). Reston: NCTM.
Zazkis, R., & Liljedahl, P. (2004). Understanding primes: The role of representation. Journal for
Research in Mathematics Education, 35, 164-186.
Zazkis, R. (2002). Language of number theory: metaphor and rigor. In Campbell, S. R., Zazkis, R.
(Eds.). Learning and teaching number theory: Research in cognition and instruction (pp. 83-96).
Westport: Ablex Publishing.
ICTMT 13
Lyon
8